![The Hamiltonian for a quantum harmonic oscillator of mass ๐ in three dimensions is](https://www.gkseries.com/blog/wp-content/uploads/2023/09/The-Hamiltonian-for-a-quantum-harmonic-oscillator-of-mass-๐-in-three-dimensions-is.jpg)
Q. The Hamiltonian for a quantum harmonic oscillator of mass ๐ in three dimensions is
![](https://www.gkseries.com/blog/wp-content/uploads/2023/09/Screenshot-763.png)
where ๐ is the angular frequency. The expectation value of ๐2 in the first excited state of the oscillator in units of
![](https://www.gkseries.com/blog/wp-content/uploads/2023/09/Screenshot-764.png)
(rounded off to one decimal place) is
Ans: 2.5
Sol:
![](https://www.gkseries.com/blog/wp-content/uploads/2023/09/Screenshot-765.png)
![](https://www.gkseries.com/blog/wp-content/uploads/2023/09/Screenshot-766.png)